Post-processing of quantum instruments

TL;DR

This paper explores how quantum instruments can be post-processed sequentially, establishing a partial order among them, and characterizes the extremal elements and their relation to POVMs for foundational and practical quantum applications.

Contribution

It introduces a formal framework for post-processing quantum instruments, characterizes the extremal elements, and links instrument post-processing to POVM transformations, advancing quantum measurement theory.

Findings

Characterized the greatest and least elements in the post-processing order.

Connected post-processing of instruments to classical outcome randomization.

Provided examples illustrating post-processing relations among different quantum instruments.

Abstract

Studying sequential measurements is of the utmost importance to both the foundational aspects of quantum theory and the practical implementations of quantum technologies, with both of these applications being abstractly described by the concatenation of quantum instruments into a sequence of certain length. In general, the choice of instrument at any given step in the sequence can be conditionally chosen based on the classical results of all preceding instruments. For two instruments in a sequence we consider the conditional second instrument as an effective way of post-processing the first instrument into a new one. This is similar to how a measurement described by a positive operator-valued measure (POVM) can be post-processed into another by way of classical randomization of its outcomes using a stochastic matrix. In this work we study the post-processing relation of instruments and the partial order it induces on their equivalence classes. We characterize the greatest and the least element of this order, give examples of post-processings between different types of instruments and draw connections between post-processings of some of these instruments and their induced POVMs.